Iceland in winter with a toddler

We are exploring a short trip to Iceland this January to see the Northern Lights and the scenery and we would be bringing a toddler and Grandma. Your suggestions would be appreciated. Organized tours may not be suitable but should we consider a private tour or can we do it independently?
What sites will be suitable and which should we avoid because of the one and half year old traveler? All suggestions appreciated. Thanks.

I think you pretty much have to stay in the southern part of Iceland because of the time of year. Many people drive the ringroad,but this is not advisable in January due to snow, road closings,etc. Maybe you could base yourself in Reykjavik at an apartment-hotel and do day trips from there. Viator.com has a 6-day private tour which might be worth investigating. Be sure to pack a swim suit, as there are lots of thermal pools around.